The first such quiz was two days ago, at the end of class. Since then, there's been some practice time, and I tell the class that I expect to see improvement from everyone. There are two versions of the quiz available to students, and they can choose the one they'll take.

Both versions of the quiz consist of six linear systems to be solved. The first has two problems each at Levels 1 through 3; the other has three each at Levels 3 and 4. By now, students are pretty familiar with the "levels," and that helps frame our conversations about the topic. Just to review:

Level 1: Two equations in which y is isolated, so students just have to set the two sides equal and solve.

Level 2: Either x or y is isolated in one equation, but not the other, so students must make a substitution step before solving.

Level 3: Two equations in standard form. At least one variable in one equation has a coefficient of 1, so it takes one step to isolate a variable. From there, it becomes a "Level 2" problem.

Level 4: Two equations in standard form. No variables have a coefficient equal to 1. (At the end of today's class, we'll start to look a little closer at another method for solving such systems.)

I give students about 20 minutes to complete this quiz. Some kids need less than that, and others need a little more. As students finish, I return their Mastery Problems from yesterday.

I've noticed that for most of the kids who struggle with this quiz, the challenge lies as much with solving linear equations as solving systems. Students understand why algebraic substitution is such an important idea (and that's a big deal), but the algebraic maneuvers might still slow them down. That's why linear systems are such a useful topic to teach, and that's why it's imperative to be willing to differentiate and meet kids where they are.

When I look at work, I'll look to see which students understand substitution, but need practice on the algebra skills (like inverse operations, use of the distributive property, and combining like terms), and which kids just don't quite get substitution yet. I'll regroup students and try to fill in the gaps accordingly during upcoming lessons.

The first problem is on the second slide of today's lesson notes. It is followed by three examples of student work on slides 3 through 5. I project the first example (slide 3) on the board and ask if anyone can find the error. This student wrote the equations correctly, but then mis-transcribed a positive sign during his substitution step. We repeat the drill for the next example (slide 4), where the student set up an equation to find the difference between the two numbers rather than their sum. The third example is correct (slide 5), and I tell the class how much I admire the visual representation of substitution I can see here, which also gives me the opportunity to to reiterate what an important idea it is, that we're able to make such a move.

It's also interesting to note the brevity of that correct example solution. It takes up less space than the other two, and at a glance, might not be as "nice-looking". Kids appreciate this observation, which helps them develop the understanding that "showing your work" doesn't necessarily mean the same thing as "writing a lot".

Problem 2: Two Candles

The work that students submitted for the candle problem was much more inconsistent than their work on the first problem. That doesn't surprise me, because it had been a few days since we've tried a problem like this, and it might have felt a little out of left-field to some kids. I also think that I should have provided a little more time to work on this problem, and that's what I did for any students who finished early on today's Mastery Quiz.

Some students tried to graph the problem, others to write equations and solve by substitution. For some, the necessary conversion from burning down "one centimeter every 20 minutes" to "3 cm per hour" was a sticking point.

Because kids are all over the place on this one, I don't spend too much time on it today. If we have time, we might write equations and sketch a graph. Just to develop understanding, I'll ask, "Which question does substitution answer: where they meet or which candle burns out first?" I want to make sure that kids are getting that concept. (From there, a neat extension question is: what would substitution say if they were never the same height?)

Otherwise, I'll save this problem for later: when we get to our review sessions next week, this will be a great place to start. We review the "Really Big Ideas" on the eighth slide, and that helps to frame the last problem we'll look at today.

With the distinction between solving systems by graphing versus by substitution laid out, it's time to introduce a third method for solving systems! I called this lesson "How Far Can Substitution Take You?" for two reasons. First, the mastery problems from the end of yesterday's class and the quiz at the start this lesson should provide evidence of how far students have come with using substitution. Second, there's a limit to the usefulness of this tool, and that's what we'll see right now.

On the ninth slide of today's lesson notes a new problem called "Pie Day!" I change the names for each class to feature two of my students, which adds to the fun. This problem makes it clear to students that there must be an alternative to graphing or substitution, both of which are pretty clunky methods here. Something like elimination (solving by addition) is going to have to exist!